As an exhaust purification apparatus for purifying exhaust gas that is exhausted by an engine, diesel particulate filters (DPF) adapted to collect and eliminate particulate matters (PM), three-way catalytic converters adapted to simultaneously purify carbon monoxide (CO), hydrocarbon (HC) and nitrogen oxide (NOx), selective catalytic reduction (SCR) converters adapted to purify NOx with use of reducing agent, and other devices have been put in use. In such an exhaust purification apparatus, the accumulation of, for instance, sulfur components contained in the exhaust gas in an exhaust purification element (e.g., DPF) provided to an exhaust pipe of the engine will cause problems such as degradation of catalysts and increase in exhaust gas pressure. Thus, as is disclosed in Japanese Laid-Open Patent Application Publication No. 2005-76495 (Patent Document 1), a proposal has been made on a technique that: estimates an accumulation amount of sulfur components accumulated in the exhaust purification element based on a fuel consumption amount and a sulfur content percentage of the fuel; and forcefully removes the sulfur components when the estimated amount is equal to or greater than a predetermined amount.